A general Schwarz Lemma for almost-Hermitian manifolds

TL;DR

This paper extends Yau's Schwarz Lemma to almost-Hermitian manifolds, providing new comparison theorems and showing restrictions on the existence of certain negatively curved metrics on product manifolds.

Contribution

It generalizes the Schwarz Lemma to almost-Hermitian manifolds and develops a Laplacian comparison theorem in this setting.

Findings

Extended Schwarz Lemma to almost-Hermitian manifolds

Proved Laplacian comparison theorem for this setting

Product of two almost-complex manifolds cannot have certain negatively curved metrics

Abstract

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions.